This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here.

Given a smooth arc (part of an ellipse actually) on the complex plane by

$z=\cos t + 0.5 i \sin t,\; t\in[\pi/10,\pi/5] $ ,

and a non-analytic function $f(z) = \text{Re } z $ defined on the arc. Obviously, $f(z) = g(t) := \cos t.$
Suppose we compute the "derivatives" of $f$ on the arc recursively by
$f'(z) = g'(t)/z'(t),\quad$ $f''(z) = \dfrac{df'(z)}{dt}\dfrac{1}{z'(t)},\quad$ $f'''(z) = \dfrac{df''(z)}{dt}\dfrac{1}{z'(t)},\quad \dots$

Is there an estimate on the upper bound of magnitude of $n^\text{th}$ order derivative of $f$ ? For example, can we show something like

$|f^{(n)}(z)|\leq C n! r^n $, where $C$ and $r$ are positive constants independent of $n$ ?

Note that in the case above the form of $f$ is really simple. If $f$ is more complicated, for example, $f\circ z(t) := \frac{|z'(t)|}{z'(t)}$, what can we say about $|f^{(n)}(z)|$ ?